Problem 71 - Algebra 2

Question

State whether each trinomial is a perfect square. If so, factor it. \(9 x^{2}-12 x+16\)

Step-by-Step Solution

Answer

The trinomial is not a perfect square and cannot be factored as such.

Step 1: Identify the Structure of a Perfect Square Trinomial

A perfect square trinomial takes the form \[a^2 - 2ab + b^2\] or \[a^2 + 2ab + b^2\].In this case, we have \(9x^2 - 12x + 16\). We'll try to match this with the structure of a perfect square.

Step 2: Check the Square Terms

The given trinomial is \(9x^2 - 12x + 16\). Identify \(a^2\) and \(b^2\). Here, \(a^2 = 9x^2\) which means \(a = 3x\) and \(b^2 = 16\) which means \(b = 4\).

Step 3: Check the Middle Term Condition

For the trinomial to be a perfect square, the middle term should be \(-2ab\) or \(+2ab\). Calculate \(-2ab = -2 \cdot 3x \cdot 4 = -24x\). The middle term \(-12x\) does match with \(-2ab\) hence it should be adjusted.

Step 4: Re-evaluate the Middle Term

The middle term from our calculation \(-24x\) doesn't match the original trinomial's middle term \(-12x\). Thus, this trinomial is not a perfect square. No perfect square factorization is possible.

Key Concepts

Trinomial Factoring

Trinomial factoring involves breaking down a trinomial into simpler components that, when multiplied, recreate the original expression. Understanding whether a trinomial is a perfect square is part of this process. If a trinomial fits the structure of a perfect square trinomial, it can be expressed in the form

• \( (a + b)^2 = a^2 + 2ab + b^2 \)
• \( (a - b)^2 = a^2 - 2ab + b^2 \)

To factor a perfect square trinomial, identify the values of \(a\) and \(b\) such that \(a^2\) and \(b^2\) form the square terms. Then, verify the middle term as \(2ab\) (or \(-2ab\) for subtraction). If it matches, factor as \((a \pm b)^2\).
However, if the middle term does not match, the trinomial cannot be factored as a perfect square. For the example \(9x^2 - 12x + 16\), the calculated middle term was \(-24x\), unlike the given \(-12x\). This indicates it is not a perfect square.

Polynomial Structure

A polynomial is a mathematical expression composed of variables, coefficients, and exponents. Its general structure is represented with terms that can be combined through addition or subtraction. Polynomials consist of one or more terms, such as monomials, binomials, and trinomials.
Trinomials, like the one in our exercise, consist of three terms and often take the form \(ax^2 + bx + c\).
Understanding the structure of a trinomial is vital before attempting any factoring. It helps in identifying patterns, especially when checking if the trinomial is a perfect square. Determining the coefficients and variable parts correctly allows for accurate manipulation and factoring attempts. In the form \(a^2 + 2ab + b^2\), identifying \(a\) and \(b\) is foundational to potentially expressing it as a square of a binomial.
Acknowledging the structure helps in recognizing patterns and the numerical logic behind polynomial expressions, facilitating a smoother factoring process.

Algebraic Expressions

Algebraic expressions are combinations of coefficients, variables, and mathematical operations (like addition, subtraction, multiplication, etc.). These expressions form the backbone of algebra and include terms such as those in our case: \(9x^2 - 12x + 16\).
Each term in an algebraic expression involves constant numbers (coefficients) and variables (like \(x\) in our example). Importantly, algebraic manipulation requires a firm grasp of each component of the expression. This includes understanding how each part interacts within operations to transform or simplify expressions.
For trinomial expressions, recognizing the role of each term is crucial in determining what kind of factoring technique to use, like in factoring perfect squares. When handling algebraic expressions, it’s essential to utilize systematic approaches to simplify and manipulate the terms. This develops strong foundational skills that are applicable across various mathematical problems and scenarios. Proper understanding and manipulation of these expressions assist greatly in problem-solving and explorational learning in mathematics.